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Mean values of multiplicative functions and applications to residue-class distribution
- Part of
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- Journal:
- Proceedings of the Edinburgh Mathematical Society , First View
- Published online by Cambridge University Press:
- 10 February 2025, pp. 1-19
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We provide a uniform bound on the partial sums of multiplicative functions under very general hypotheses. As an application, we give a nearly optimal estimate for the count of
$n \le x$ for which the Alladi–Erdős function 
$A(n) = \sum_{p^k \parallel n} k p$ takes values in a given residue class modulo q, where q varies uniformly up to a fixed power of 
$\log x$. We establish a similar result for the equidistribution of the Euler totient function 
$\phi(n)$ among the coprime residues to the ‘correct’ moduli q that vary uniformly in a similar range and also quantify the failure of equidistribution of the values of 
$\phi(n)$ among the coprime residue classes to the ‘incorrect’ moduli.
